document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209633>Question 1209633</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all complex numbers z such that |z - 1| = |z + 3| + |z - 2i|.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849731 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849731\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to approach this problem:\r<BR>\n" );
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document.write( "1. **Geometric Interpretation:**\r<BR>\n" );
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document.write( "The equation |z - 1| represents the distance between the complex number z and the complex number 1 (which can be thought of as the point (1,0) in the complex plane).\r<BR>\n" );
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document.write( "|z + 3| represents the distance between z and -3 (or (-3,0)).\r<BR>\n" );
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document.write( "|z - 2i| represents the distance between z and 2i (or (0,2)).\r<BR>\n" );
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document.write( "The equation states that the distance from z to 1 is equal to the sum of the distances from z to -3 and z to 2i.\r<BR>\n" );
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document.write( "2. **Consider the Triangle Inequality:**\r<BR>\n" );
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document.write( "The triangle inequality states that for any complex numbers a, b, and c:\r<BR>\n" );
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document.write( "|a + b| ≤ |a| + |b|\r<BR>\n" );
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document.write( "Let a = z + 3 and b = z - 2i, then |z + 3 + z - 2i| ≤ |z+3| + |z-2i|<BR>\n" );
document.write( "|2z + 3 - 2i| ≤ |z+3| + |z-2i|\r<BR>\n" );
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document.write( "We are given |z - 1| = |z + 3| + |z - 2i|.<BR>\n" );
document.write( "If z were on the line segment joining -3 and 2i, then |z+3| + |z-2i| would equal the distance between -3 and 2i. In this case, |z-1| = |-3-2i| = √13. But z would be a real number between -3 and 0, or an imaginary number between 0 and 2i. Neither of these satisfy the equation.\r<BR>\n" );
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document.write( "3. **Try z = x + yi:**\r<BR>\n" );
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document.write( "Substitute z = x + yi into the equation:\r<BR>\n" );
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document.write( "|x + yi - 1| = |x + yi + 3| + |x + yi - 2i|<BR>\n" );
document.write( "|(x - 1) + yi| = |(x + 3) + yi| + |x + (y - 2)i|<BR>\n" );
document.write( "√((x - 1)² + y²) = √((x + 3)² + y²) + √(x² + (y - 2)²)\r<BR>\n" );
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document.write( "4. **Square both sides (carefully!):**\r<BR>\n" );
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document.write( "(x - 1)² + y² = (x + 3)² + y² + x² + (y - 2)² + 2√((x + 3)² + y²)(x² + (y - 2)²)<BR>\n" );
document.write( "x² - 2x + 1 + y² = x² + 6x + 9 + y² + x² + y² - 4y + 4 + 2√((x + 3)² + y²)(x² + (y - 2)²)<BR>\n" );
document.write( "-2x + 1 = 6x + 13 + x² + y² - 4y + 2√((x + 3)² + y²)(x² + (y - 2)²)<BR>\n" );
document.write( "-8x - 12 = x² + y² - 4y + 2√((x + 3)² + y²)(x² + (y - 2)²)\r<BR>\n" );
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document.write( "This equation is getting very complicated.\r<BR>\n" );
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document.write( "5. **Consider specific cases:**\r<BR>\n" );
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document.write( "Since the geometric interpretation is difficult, and the algebraic substitution is messy, there may not be a simple solution.\r<BR>\n" );
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document.write( "If we let y=0, we have |x-1| = |x+3| + |x|.<BR>\n" );
document.write( "If x > 1, x-1 = x+3+x which is impossible.<BR>\n" );
document.write( "If -3< x < 1, 1-x = x+3+x which is impossible.<BR>\n" );
document.write( "If x < -3, 1-x = -x-3-x so 1 = -3-x, and x=-4.<BR>\n" );
document.write( "So z=-4 is one solution.\r<BR>\n" );
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document.write( "Since the algebra is so difficult, and we found one solution, it is possible there are no others.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "z = -4 is one solution.  The presence of the square roots makes it very difficult to solve for other solutions algebraically.  It's possible this is the only solution.<BR>\n" );
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